Green’s Dictionary of Slang

1. sleepy, peaceful.

1738		Ordinary of Newgate Account 18 Jan. n.p.: He followed picking of Pockets, and other trivial Robberies, and if he met with a Dossy-Cull (that is, a Man asleep) he bit him, (that is, robb'd him.).
1738		Ordinary of Newgate Account 8 Nov. n.p.: ] They [...] told her they had brought her a Daucy Cock, (a sleeping Cull) who had 50 or 60 Ridges (Guineas) about him.
1881		Newcastle Courant 2 Sept. 6/5: Had it been in a busy district his fakement would have been no go but it was a quiet dossy neighbourhood.

2. ineffectual, weak, ‘soft’.

1958

(con. 1940s) B. Behan Borstal Boy 264: Don’t know what she ever saw in that dossy bastard.